A Note on k-Wise Oddtown Problems

Abstract

For integers $2 \le t \le k$, we consider a collection of k set families ${\mathcal {A}}_j: 1 \le j \le k$ where ${\mathcal {A}}_j = \{ A_{j,i} \subseteq [n] : 1 \le i \le m \}$ and $|A_{1, i_1} \cap \cdots \cap A_{k,i_k}|$ is even if and only if at least t of the $i_j$ are distinct. In this paper, we prove that $m =O(n^{ 1/ \lfloor k/2 \rfloor })$ when $t=k$ and $m=O(n^{1/(t-1)})$ when $2t-2 \le k$ and prove that both of these bounds are best possible. Specializing to the case where ${\mathcal {A}}= {\mathcal {A}}_1 = \cdots = {\mathcal {A}}_k$, we recover a variation of the classical oddtown problem.